10: Evolutionary Games

1. 10: Evolutionary Games Games People Play.

2. Evolutionary Games • What if individuals aren’t as smart and calculating as we have assumed so far? • Perhaps decision making is simpler • Good decisions and decision makers persist and are copied • Bad decisions and decision makers die out!! • Natural selection makes the decisions • This is the idea behind evolutionary game theory Games People Play.

3. Evolutionary Games • Genotype – the genetic type of a player • Phenotype – the behavior of a genotype • Fitness – a measure of the success of a phenotype • Selection – successful genotypes out-reproduce unsuccessful ones • Mutations – random creation of new genotypes • Invasion – mutations that successfully out-compete the current genotypes and increase in number • Evolutionary stability – a population of genotypes that cannot be successfully invaded Games People Play.

4. The prisoners dilemma as an evolutionary game • 2 genotypes • Cooperators (C-types) – always cooperate • Defectors (D-types) - always defect • Pairs of players are matched at random • A cooperator can be matched with another cooperator or with a defector and vice versa. Games People Play.

5. The prisoners dilemma as an evolutionary game • The payoff matrix • Suppose that the proportions of cooperators and defectors in the population are initially x and 1-x respectively. Games People Play.

6. The prisoners dilemma as an evolutionary game • Fitness levels • A cooperator meets another cooperator with probability x and a defector with probability 1-x and expects to earn F(c) = x(12) + (1-x)(1) • A defector will also meet a cooperator with probability x and a defector with probability 1-x and expects to earn F(d) = x(25) + (1-x)(3) Games People Play.

7. The prisoners dilemma as an evolutionary game • Selection • The cooperators will outbreed the defectors if F(C) > F(D) x(12) + (1-x)(1) > x(25) + (1-x)(3) • So the cooperators will outbreed the defectors if x < - (2/11) • Which cannot hold. • So the cooperators will die out!!! • 100% defectors is an Evolutionary Stable State. Games People Play.

8. The prisoners dilemma as an evolutionary game • Mutation • Suppose now a mutation occurs and a third genotype that plays tit-for-tat appears. • Also suppose that each pair of players plays each other three times. • Assume a T-type always plays cooperate on the first round. • Can the mutation successfully invade? Games People Play.

9. The prisoners dilemma as an evolutionary game • If a defector meets a defector we get • Round #1 • Round #2 • Round #3 Games People Play.

10. The prisoners dilemma as an evolutionary game • So each defector that meets another defector enjoys a fitness of 9 Games People Play.

11. The prisoners dilemma as an evolutionary game • If a defector meets a tit-for-tat we get • Round #1 • Round #2 • Round #3 Games People Play.

12. The prisoners dilemma as an evolutionary game • If a defector meets a tit-for-tat • The tit-for-tat enjoys a fitness of 7 • The Defector enjoys a fitness of 31 Games People Play.

13. The prisoners dilemma as an evolutionary game • If a tit-for-tat meets a tit-for-tat we get • Round #1 • Round #2 • Round #3 Games People Play.

14. The prisoners dilemma as an evolutionary game • If a tit-for-tat meets a tit-for-tat • Both tit-for-tats enjoy a fitness of 36 Games People Play.

15. The prisoners dilemma as an evolutionary game • So we know • Defector meets defector • Both receive a fitness of 9 • Defector meets a tit-for-tat • Defector receives a fitness of 31 • Tit-for-tat receives a fitness of 7 • Tit-for-tat meets a tit-for-tat • Both receive a fitness of 36. Games People Play.

16. The prisoners dilemma as an evolutionary game • Conclusions • There are two evolutionary stable steady states • A tit-for-tat cannot invade a population of defectors • A defector cannot invade a population of tit-for-tats • If both types initially exist in the population which ESS arises depends on their initial relative numbers. • Historymatters Games People Play.